# American Institute of Mathematical Sciences

February  2021, 26(2): 795-813. doi: 10.3934/dcdsb.2020142

## Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

 1 College of of Sciences, Northeastern University, Shenyang 110819, P. R. China 2 School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, P. R. China

* Corresponding author: Xiaopeng Zhao

Received  January 2018 Revised  January 2020 Published  February 2021 Early access  May 2020

Fund Project: This paper is supported by the National Nature Science Foundation of China (grant No. 11401258), Nature Science Foundation of Jiangsu Province (grant No. BK20140130) and China Postdoctoral Science Foundation (grant No. 2015M581689)

The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all $t>0$ step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the $L^p$ decay estimate for the solution $u(\cdot,t)$ and all its derivatives for generalized Navier-Stokes equations with $\max\{1,\frac{3+q}6\}<\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\}$.

Citation: Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604. [2] D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations,, [Contributions to Current Challenges in Mathematical Fluid Mechanics], in Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31–51. [3] P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.  doi: 10.1137/S0036141098337333. [4] X. Ding and J. Wang, Global solution for a semilinear parabolic system, Acta Math. Sci. (English Ed.), 3 (1983), 397-414.  doi: 10.1016/S0252-9602(18)30621-0. [5] J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.  doi: 10.3934/krm.2013.6.545. [6] A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969. [7] D. Hoff and J. A. Smooler, Global existence for systems of parabolic conservation laws in several space variables, J. Differential Equations, 68 (1987), 210-220.  doi: 10.1016/0022-0396(87)90192-6. [8] D. Hoff and J. A. Smooler, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 213-235.  doi: 10.1016/S0294-1449(16)30403-6. [9] Z. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.  doi: 10.1016/j.na.2012.04.014. [10] Z. Jiang and J. Fan, Time decay rate for two 3D magnetohydrodynamics-$\alpha$ models, Math. Methods Appl. Sci., 37 (2014), 838-845.  doi: 10.1002/mma.2840. [11] Q. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptot. Anal., 94 (2015), 105-124.  doi: 10.3233/ASY-151307. [12] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182. [13] I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.  doi: 10.1512/iumj.2001.50.2084. [14] I. Kukavica and J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.  doi: 10.1088/0951-7715/19/2/003. [15] N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyperdissipation, Geom. Funct. Anal., 12 (2002), 355-379.  doi: 10.1007/s00039-002-8250-z. [16] H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. Zingano, Decay in time of incompressible flows, J. Math. Fluid Mech., 5 (2003), 231-244.  doi: 10.1007/s00021-003-0079-1. [17] J. Leray, Étude de diverses équations integrales non lineaires et de quelques problémes que pose l'hydrodynamique, Thèses de l'entre-deux-guerres, 142 (1933), 88pp. [18] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [19] P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.  doi: 10.1016/j.jfa.2010.07.013. [20] J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. [21] Q. Liu and J. Zhao, Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces, J. Math. Anal. Appl., 420 (2014), 1301-1315.  doi: 10.1016/j.jmaa.2014.06.031. [22] Q. Liu, J. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl. (4), 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8. [23] S. Liu, F. Wang and H. Zhao, Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.  doi: 10.1016/j.jde.2007.02.014. [24] C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.  doi: 10.1016/j.na.2006.11.011. [25] M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111. [26] M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443. [27] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, USA, 1970. [28] W. A. Strauss, Decay and asymptotic for $u_tt-\Delta u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.  doi: 10.1016/0022-1236(68)90004-9. [29] S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Method Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868. [30] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\mathbb{R}^n$, J. London Math. Soc., 35 (1987), 303-313.  doi: 10.1112/jlms/s2-35.2.303. [31] J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007. [32] J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530. [33] Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x. [34] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014. [35] Y. Zhou, A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.  doi: 10.1002/mma.841. [36] Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.  doi: 10.1088/0951-7715/21/9/008.

show all references

##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604. [2] D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations,, [Contributions to Current Challenges in Mathematical Fluid Mechanics], in Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31–51. [3] P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.  doi: 10.1137/S0036141098337333. [4] X. Ding and J. Wang, Global solution for a semilinear parabolic system, Acta Math. Sci. (English Ed.), 3 (1983), 397-414.  doi: 10.1016/S0252-9602(18)30621-0. [5] J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.  doi: 10.3934/krm.2013.6.545. [6] A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969. [7] D. Hoff and J. A. Smooler, Global existence for systems of parabolic conservation laws in several space variables, J. Differential Equations, 68 (1987), 210-220.  doi: 10.1016/0022-0396(87)90192-6. [8] D. Hoff and J. A. Smooler, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 213-235.  doi: 10.1016/S0294-1449(16)30403-6. [9] Z. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.  doi: 10.1016/j.na.2012.04.014. [10] Z. Jiang and J. Fan, Time decay rate for two 3D magnetohydrodynamics-$\alpha$ models, Math. Methods Appl. Sci., 37 (2014), 838-845.  doi: 10.1002/mma.2840. [11] Q. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptot. Anal., 94 (2015), 105-124.  doi: 10.3233/ASY-151307. [12] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182. [13] I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.  doi: 10.1512/iumj.2001.50.2084. [14] I. Kukavica and J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.  doi: 10.1088/0951-7715/19/2/003. [15] N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyperdissipation, Geom. Funct. Anal., 12 (2002), 355-379.  doi: 10.1007/s00039-002-8250-z. [16] H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. Zingano, Decay in time of incompressible flows, J. Math. Fluid Mech., 5 (2003), 231-244.  doi: 10.1007/s00021-003-0079-1. [17] J. Leray, Étude de diverses équations integrales non lineaires et de quelques problémes que pose l'hydrodynamique, Thèses de l'entre-deux-guerres, 142 (1933), 88pp. [18] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [19] P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.  doi: 10.1016/j.jfa.2010.07.013. [20] J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. [21] Q. Liu and J. Zhao, Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces, J. Math. Anal. Appl., 420 (2014), 1301-1315.  doi: 10.1016/j.jmaa.2014.06.031. [22] Q. Liu, J. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl. (4), 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8. [23] S. Liu, F. Wang and H. Zhao, Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.  doi: 10.1016/j.jde.2007.02.014. [24] C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.  doi: 10.1016/j.na.2006.11.011. [25] M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111. [26] M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443. [27] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, USA, 1970. [28] W. A. Strauss, Decay and asymptotic for $u_tt-\Delta u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.  doi: 10.1016/0022-1236(68)90004-9. [29] S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Method Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868. [30] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\mathbb{R}^n$, J. London Math. Soc., 35 (1987), 303-313.  doi: 10.1112/jlms/s2-35.2.303. [31] J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007. [32] J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530. [33] Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x. [34] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014. [35] Y. Zhou, A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.  doi: 10.1002/mma.841. [36] Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.  doi: 10.1088/0951-7715/21/9/008.
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